A Hybrid Symbolic-Numeric Method for Multiple Integration Based on Tensor-Product Series Approximations

Abstract

This work presents a new hybrid symbolic-numeric method for fast and accurate evaluation of multiple integrals, effective both in high dimensions and with high accuracy. In two dimensions, the thesis presents an adaptive two-phase algorithm for double integration of continuous functions over general regions using Frederick W. Chapman's recently developed Geddes series expansions to approximate the integrand. These results are extended to higher dimensions using a novel Deconstruction/Approximation/Reconstruction Technique (DART), which facilitates the dimensional reduction of families of integrands with special structure over hyperrectangular regions. The thesis describes a Maple implementation of these new methods and presents empirical results and conclusions from extensive testing. Various alternatives for implementation are discussed, and the new methods are compared with existing numerical and symbolic methods for multiple integration. The thesis concludes that for some frequently encountered families of integrands, DART breaks the curse of dimensionality that afflicts numerical integration.